Multiplication Methods A collection of some methods for doing multiplication that have been devised and used in previous times.
Notes on Multiplication Methods ~ 1 All of the methods given here are written methods. For an additional account of some mechanical methods see under 'Artifacts for Mathematics' (in the trol menu) where the material for making, together with instructions for using: Napier's Rods, Genaille's Rods and a Slide Rule, are to be found. At the back of this unit are some sheets of multiplication tables. The first is simply eight copies of a basic 1 to 9 multiplication square, which makes it easy to provide everyone with their own personal copy when those facts are needed. Needing to know their tables should not be made an issue in this work. The second, A Book of Multiplication Tables is much more complicated to make. First of all it requires some care with the double-sided photocopying needed. Check that the crosses in the middle of the two sheets match (within 2 mm if possible), and also that the 9 table does back the 9 table (and NOT the 65 ). The sheet has then to be folded in half three times. Do not rush this stage. Press the folds down well with some hard object (ruler or pen). The dotted lines are only a guide, the important thing is to match up the edges of the paper each time, and see that the dotted lines are on the inside of the fold. The titled front cover should always be on the outside of the fold. A staple through the last fold (the hinged edge) and three cuts along the outer edges with a guillotine will complete the process. It is nowhere near as difficult as it sounds! An alternative cover for the booklet is provided. This will have to be cut and pasted onto the master copy before the photocopying is done. All of this would only be worthwhile if the tables might be used in some other work. These same tables are also available in a straight sheet format if wanted. They can be found in the Division unit (in the trol menu). It must be emphasied that there are many more ways of doing multiplication than those given here, though most of them are merely variations on a general theme. The ones suggested in this unit are those considered to offer the most useful insights into the historic development of this particular piece of arithmetic so some might appreciate that No, it hasn t always been like that! Exactly how any or all of this material is used will be, as always, a matter for individual teachers to decide and plan for. It can be surprising to see the lengths to which mathematicians and other professional users of mathematics have gone to over the centuries to simplify the multiplication process. But remember that even so great a mathematician as Leibnitz (166 to 1716) who invented the calculus said It is unworthy of excellent men to lose hours like slaves in the labour of calculation. (A feeling which many of our pupils share!) Frank Tapson 200 [trolfg:2]
Using Doubling and Halving Notes on Multiplication Methods ~ 2 1. The Ancient Egyptian Method This is probably the oldest of all the written methods. It is of some interest because of its links with the binary form of a number, and also because of the simplicity of the idea - it needs only an ability to be able to multiply by 2 and add. However it should also be appreciated that in those days (1000 BC or so) such things were not for 'ordinary folk' who would have had no need of such a skill. It was very much the preserve of professionals such as astrologers, architects and mathematicians. 2. The Russian Peasant Method This method was more widely used than its name might imply. It was in fact taught and practised throughout Europe for quite some time. It is a less demanding algorithm than the previous one since the numbers to be crossed off are easier to identify. It is interesting to write them side by side and see how they are doing the same thing. The link of course is the binary system underlying both. 1 26 2 56 1072 21 16 2 2 576 1152 26 21 56 10 1072 5 21 2 2 1 576 1152 Using the Gelosia Method (Pronounced jee - low - see - uh) Any analysis or explanation of why this method works is probably best linked to looking at the place values of the different figures involved. The worked example from the sheet is shown on the right. 2 6 2 2 6 1 2 The lower diagram replaces each of the the figures in the top diagram with its place value, using the usual notation of H T U, with the addition of Th for thousands. The neat way each place value falls into line along the diagonals makes it very clear why the method works. It also makes it very apparent why zeros must be included in the numbers when necessary. It is easy to extend the diagram to cover tens of thousands and upwards or, tenths, hundredths, thousandths and so on downwards, and thus show why placing the decimal point works as it does. Frank Tapson 200 [trolfg:] H T U Th H Th H T H T H T U This method almost certainly originated in India, as did so much of our arithmetic. From there it spread outwards to China, Persia and the Arab world. It reached Europe through Italy where it first appeared in the 1th century. The name 'gelosia' was given to it in Italy since the necessary grid resembled the lattices or gratings (= gelosia) which were made in metal and fastened in front of the windows of the houses for security. It has been claimed that Napier got the idea for his rods from this method. Whether or not he did, the possibility can clearly be seen. In fact, multiplication tables and a knowledge of place value could be dispensed with by using the two methods together. Once the grid has been drawn, a set of Napier's Rods would provide all the needed answers to complete it, and the final addition would be the only arithmetic necessary. One practical point. Do not slow things up by drawing precisely ruled grids. It is quite sufficient to write down the two numbers (adequately spaced) and sketch the grid around them free-hand. T U
Using Quarter Squares Notes on Multiplication Methods ~ The reason why this method works can only be justified by the use of algebra. It can be shown that (a + b) 2 (a b) 2 ab Given a and b are the two numbers to be multiplied together, and the two terms on the right are quarter squares (as defined on the sheet) then the algorithm follows from the above identity. And (provided the requisite tables are available) multiplication has been replaced by one addition and two subtractions without need for any multiplication facts to be known or found. In passing it is of interest to consider (and prove) why the remainder can be disregarded. Note that this method is completely accurate in giving all the figures possible in the final answer. Its big limitations are that (a) a set of tables is needed and, (b) the size of the numbers it can handle is constrained by the size of the tables (which has to cover the sum of the two numbers being multiplied). The second limitation is not too bad. The sample sheet included here goes to 600, so it is easy to visualize the smallish book which would be needed to reach 100,000. That means an accuracy of significant figures (or better) which should cover most practical needs. Approximations allow the tables to be used for larger numbers. The method was used (but how widely is not known) and the necessary tables to enable this were published in the late 100 s. Using Logarithms This is a much simplified version of what was once taught in the classrooms of every secondary school. The words characteristic and mantissa do not appear, and only numbers greater than 1 are dealt with to avoid that very troublesome negative characteristic. Also only -figure tables are used and the customary difference table does not appear. However, the essentials required to give a glimpse of what was done are there. In modern terms this could be called Logarithms Lite. The characteristic which is so necessary to control the place value has been replaced here by the e-notation. This is the notation introduced by the IEEE (= Institute of Electrical and Electronic Engineers) and used in all computer work. Gaining an acquaintance with this notation might be regarded as a valuable by-product of this piece of work. One of the inadequancies of -figure tables is very apparent once the numbers go beyond 5.0 when many of the logarithmic values are repeated. This cannot be avoided without moving on to or 5-figure tables. Sheet 1 does not require any knowledge of the characteristic as the examples are chosen so as not to go beyond the range of the table. Also there are no ambiguous values for the logarithms. Sheet 2 introduces the idea of the e-notation. Some help might be needed with the business of using the tables, first in finding the logarithm from the number, and then the much more difficult matter of reversing that process. Further work could be done with numbers less than 1 and division. Frank Tapson 200 [trolfg:]
Using Triangle Numbers Notes on Multiplication Methods ~ This method has the big advantage that it can be easily explained or justified diagrammatically, without any recourse to algebra, except for the generalisation at the end. For this purpose it is better to change the conventionally drawn triangle numbers (as equilateral triangles) into rightangled triangles. The appropriate drawing for T is shown on the right. We will do the sum using five diagrams. 1. The sum requires the number of objects making this rectangle to be counted. 2. First make the triangle number which matches the longest edge. In this case it is T. Cut away the triangle (shown in black) which is above the size of the rectangle. In this case it is T 5 5. Triangle removed was T 5 or T ( - ) leaving this incomplete rectangle 5. Complete the rectangle with smallest possible triangle (shown in black). In this case T 2 or T ( - 1) Physically we have shown that = 6 15 + = T T 5 + T 2 = T T ( ) + T ( 1) Which generalises to a b = T a T (a b) + T (b 1) Rearranged for the algorithm a b = T a + T (b 1) T (a b) Another approach to using triangle numbers is implied by this diagram which leads to a different algorithm. b a a + b b a + b a a b A similar thing can be done using square numbers (or indeed any polygon numbers). The identity (a + b) 2 = a 2 + b 2 + 2ab which is illustrated on the left can be rearranged to give ab = and the required algorithm can be produced from that. A table giving the values of square numbers S n is available. Though interesting, and illuminating, to see how these work they were never more than mathematical curiosities, and (unlike the method of quarter squares) there appears to be no evidence that they were ever used in any practical way. One advantage this method does have over quarter squares is that, in this case the largest number in the tables is also the largest possible multiplier. Frank Tapson 200 [trolfg:5]
Multiplication using Doubling and Halving 1. The Ancient Egyptian Method Consider the sum 26 First write out two columns of numbers side by side. The left-hand column starts with 1. The right-hand colum starts with the larger of the two numbers being multiplied (26). The columns are formed by doubling each of the previous numbers, stopping only when the figure in the left-hand column would become bigger than (or equal to) the other number being multiplied (). The final result would be as shown on the right. Take care to keep the columns lined up, both vertically and horizontally. The next stage is to look at the left-hand column and find the numbers which are needed to add up to the value of the smaller multiplier (). In this case the necessary set is 1 + 2 + + 2 = (Note that the bottom number should always be needed, and it is best working backwards from that, subtracting as you go.) Cross out all the numbers in that column not needed together with the numbers in the right-hand column which are on the same level. Finally, add up the numbers in the right-hand column which have not been crossed out. That is the answer. The completed sum is shown on the right. So, 26 = 1152 1 26 2 56 1072 21 16 2 2 576 1 26 2 56 1072 21 16 2 2 576 1152 Try these, using the Ancient Egyptian method. 26 1 1 61 71 21 55 97 2 10 251 21 2. The Russian Peasant Method Consider the sum 26 First write out two columns of numbers side by side. Each column starts with one of the two numbers to be multiplied. ( and 26) Make one column by doubling each of the previous numbers. Make the other column by halving each of the previous numbers, ignoring any remainders, and stopping at 1. (It is best to make the halving column under the smaller of the two multipliers) The final result would be as shown on the right. Take care to keep the columns lined up, both vertically and horizontally. The next stage is to look at the halving column and find all the even numbers. In this case they are 10 and 2 Cross out the even numbers in that column together with the numbers in the right-hand column which are on the same level. Finally, add up the numbers in the right-hand column which have not been crossed out. That is the answer. The completed sum is shown on the right. So, 26 = 1152 26 21 56 10 1072 5 21 2 2 1 576 26 21 56 10 1072 5 21 2 2 1 576 1152 Try these, using the Russian Peasant method. 2 16 0 5 6 1 12 21 71 27 5 Frank Tapson 200 [trolfg:6]
Multiplication using the Gelosia Method Consider the sum 26 (That is digits 2 digits) Sketch a grid like that on the right, and place the two numbers as shown. Next, do the six single-digit multiplications formed by matching the separate numbers across the top with those on the right-hand edge. In this case, it means doing the six sums shown on the right and the answers to those multiplications are written in the appropriate boxes of the grid for each number-pair. No answer can have more than two digits. (tens and units) Any tens (L.H. digit) are written above the diagonal line, The units (R.H. digit) are written below the diagonal line. (It may be zero) The completed grid should look like that on the right. 2 6 2 6 2 6 2 6 2 2 6 1 2 The final answer is produced by adding (the results of the multiplication only) along the diagonals, working from right to left, and carrying figures as necessary, and this is shown on the right. 2 6 2 2 6 1 2 1 1 5 2 So, 26 = 1152 Try these, using the gelosia method. 6 2 75 76 9 02 1 999 7 7 219 Decimal points are dealt with very simply. Consider the sum 2.6. Work exactly as before with 26 and then put the two decimal points in those numbers on the vertical and horizontal lines respectively. Follow those two lines into the grid until they meet, and then, moving down the diagonal line from there, put the decimal point in the answer. All of this can be seen in the diagram on the right by following the arrows. So, 2.6. = 11.52 2 6 2 2 6 1 2 1 1. 5 2 Try these, using the gelosia method. First, by using the grids already created in the previous work, write down the answers to these:.6 2 75.. 7.6 9 0.2 1 9.99 7 7. 2.19 and then do these: 12..6 7.2 1 50.7 0.617. 0.059 2 0.07 0.0 Frank Tapson 200 [trolfg:7]
Multiplication using Quarter Squares The sheet headed Values of Quarter Squares Q n is needed The quarter square of a number, is the value of that number multiplied by itself and then divided by, ignoring any remainder (if there is one). Examples The quarter square of 2 is: 2 2 = = 1 is: = 9 = 2 rem 1 which becomes 2 5 is: 5 5 = 25 = 6 rem 1 which becomes 6 and so on. Quarter squares would be of little use for the purpose of simplifying multiplication if they had to be worked out each time they were needed and so a table is used to look up their values. Look at the sheet: Values of Quarter Squares and confirm that the quarter square of 1 is 2 62 (or 2,62 if preferred). Having such a table enables us to change a multiplication sum into a much simpler matter of addition and subtraction. Consider the sum 26 First add them: 26 + = 11 and find the quarter square of that = 2 10 Then subtract them: 26 = 255 and find the quarter square of that = 12 656 Finally, subtract the quarter squares to get the final answer = 11 52 and 26 = 11 52 It could be set out something like this: 26 Q n + 11 2 10 225 12 656 11 52 Try these, using the quarter squares method. 25 2 26 5 6 12 7 6 76 21 157 Frank Tapson 200 [trolfg:]
Multiplication using Triangle Numbers The sheet headed Values of Triangle Numbers T n is needed T 1 = 1 T 2 = T = 6 T = 10 T 5 = 15 Triangle numbers are those that can be made in the way shown above where the first five are drawn. The actual value of any triangle number can be worked out using T n = n (n + 1) 2 Example The value of the triangle number for is: ( + 1) 2 = 9 2 = 6 Triangle Numbers would be of little use for the purpose of simplifying multiplication if they had to be worked out each time they were needed and so a table is used to look up their values. Look at the sheet: Values of Triangle Numbers and confirm that the triangle number corresponding to 1 is 5 91 (or 5,91 if preferred). Having such a table enables us to change a multiplication sum into a much simpler matter of addition and subtraction. Consider the sum 26 Find the triangle number corresponding to the larger (26) = 6 06 Subtract 1 from the smaller ( - 1 = 2) and find its triangle number = 12 656 Add those two triangle numbers = 6 99 Subtract the two multiplying numbers (26 - = 225) and find the triangle number for that = 25 25 Subtract that from the previous result to get the final answer = 11 52 and 26 = 11 52 It could be set out something like this: T n 26 6 06 1 = 2 90 + 6 99 225 25 25 11 52 Try these, using the triangle numbers method. 21 7 67 2 179 5 5 12 59 1 259 Frank Tapson 200 [trolfg:9]
Multiplication using Logarithms ~ 1 The sheet headed Values of Logarithms L n is needed We will not concern ourselves here with what logarithms are, but only how to use them for multiplication. (They can also be used for division.) First of all, every number has a logarithm associated with it and that association is unique. In other words, given any number it has only one logarithm and, just as importantly, given any logarithm it can only represent one number. We will start with a very simple example of how we use logarithms. Consider the sum 2 To use the table of logarithms we need to see this as 2.00.00 From the table we can find that 2.00 has a logarithm of 0.01 and that.00 has a logarithm of 0.602 Then we ADD the two logarithms to get 0.90 That is the answer to the sum but, it is a logarithm, so we need to find what number it represents. Using the table (in reverse) shows that 0.90 is the logarithm of.00 So 2.00.00 =.00 (Hardly a surprise, but good to see it works.) Now consider the much harder sum 2.15.62 We find their logarithms are 0.2 and 0.559 and these add to 0.91 The logarithm 0.91 matches the number 7.7 So 2.15.62 = 7.7 It could be set out something like this: N L n 2.15 0.2.62 0.559 + 7.7 0.91 The full answer should have been 7.7 but it must be recognised that the very simplified table we are using is not capable of that sort of accuracy. We wish only to establish how logarithms are used and, for most practical purposes, this accuracy is adequate. Try doing these, using logarithms 2 1.92.6 5.1 1.5.55 1.29.07 1.9 Frank Tapson 200 [trolfg:10]
Multiplication using Logarithms ~ 2 Notice that the table of logarithms covers only numbers from 1.00 to 9.99 and that is true for all tables of logarithms. For greater accuracy it might be 1.0000 to 9.9999 but the principle is the same. So now we consider how to adapt all numbers to fit the tables and see how powerful this tool is in handling all multplications. For this we need to be familiar with a particular way of writing numbers which first sets them down as a value between 1 and 10 and then puts enough 10's after them so they would be restored to their correct value. For example can be written as. 10 26 can be written as 2.6 10 10 There is a special notation used for this which writes the above numbers as.e1 and 2.6e2 respectively where the number after the 'e' shows how many 10's are needed. Or it can be thought of as the amount that the decimal point must be moved to the right to restore the correct value. Now when we look up the logarithms of. and 2.6 we replace the 0 on the front with the e-number (if there is one). So that the logarithm of (log. = 0.6 with e = 1) becomes 1.6 and the logarithm of 26 (log 2.6 = 0.2 with e = 2) becomes 2.2 Then we add the two logarithms to get.061 Remembering this stands for a logarithm of 0.061 with e = We next find 0.061 in the logarithm tables and see that it corresponds to the number 1.15 With e = we must multiply by 10, times 1.15 10 10 10 10 = 11500 More accurately 26 = 1152 But we can write 26 = 11500 (to significant figures) It could be set out something like this: N L n 1.6 26 2.2 + 1.15.061 Try doing these, using logarithms 7 26 5 7 27 6 517 2 65. 17 610 90 Frank Tapson 200 [trolfg:11]
Values of Quarter Squares Q n for n = 1 to 600 1 0 51 650 101 2550 151 5700 201 10 100 251 15 750 2 1 52 676 102 2601 152 5776 202 10 201 252 15 76 2 5 702 10 2652 15 552 20 10 02 25 16 002 5 729 10 270 15 5929 20 10 0 25 16 129 5 6 55 756 105 2756 155 6006 205 10 506 255 16 256 6 9 56 7 106 209 156 60 206 10 609 256 16 7 12 57 12 107 262 157 6162 207 10 712 257 16 512 16 5 1 10 2916 15 621 20 10 16 25 16 61 9 20 59 70 109 2970 159 620 209 10 920 259 16 770 10 25 60 900 110 025 160 600 210 11 025 260 16 900 11 0 61 90 111 00 161 60 211 11 10 261 17 00 12 6 62 961 112 16 162 6561 212 11 26 262 17 161 1 2 6 992 11 192 16 662 21 11 2 26 17 292 1 9 6 102 11 29 16 672 21 11 9 26 17 2 15 56 65 1056 115 06 165 606 215 11 556 265 17 556 16 6 66 109 116 6 166 69 216 11 66 266 17 69 17 72 67 1122 117 22 167 6972 217 11 772 267 17 22 1 1 6 1156 11 1 16 7056 21 11 1 26 17 956 19 90 69 1190 119 50 169 710 219 11 990 269 1 090 20 100 70 1225 120 600 170 7225 220 12 100 270 1 225 21 110 71 1260 121 660 171 710 221 12 210 271 1 60 22 121 72 1296 122 721 172 796 222 12 21 272 1 96 2 12 7 12 12 72 17 72 22 12 2 27 1 62 2 1 7 169 12 17 7569 22 12 5 27 1 769 25 156 75 106 125 906 175 7656 225 12 656 275 1 906 26 169 76 1 126 969 176 77 226 12 769 276 19 0 27 12 77 12 127 02 177 72 227 12 2 277 19 12 2 196 7 1521 12 096 17 7921 22 12 996 27 19 21 29 210 79 1560 129 160 179 010 229 1 110 279 19 60 0 225 0 1600 10 225 10 100 20 1 225 20 19 600 1 20 1 160 11 290 11 190 21 1 0 21 19 70 2 256 2 161 12 56 12 21 22 1 56 22 19 1 272 1722 1 22 1 72 2 1 572 2 20 022 29 176 1 9 1 6 2 1 69 2 20 16 5 06 5 106 15 556 15 556 25 1 06 25 20 06 6 2 6 19 16 62 16 69 26 1 92 26 20 9 7 2 7 192 17 692 17 72 27 1 02 27 20 592 61 196 1 761 1 6 2 1 161 2 20 76 9 0 9 190 19 0 19 90 29 1 20 29 20 0 0 00 90 2025 10 900 190 9025 20 1 00 290 21 025 1 20 91 2070 11 970 191 9120 21 1 520 291 21 170 2 1 92 2116 12 501 192 9216 22 1 61 292 21 16 62 9 2162 1 5112 19 912 2 1 762 29 21 62 9 2209 1 51 19 909 2 1 29 21 609 5 506 95 2256 15 5256 195 9506 25 15 006 295 21 756 6 529 96 20 16 529 196 960 26 15 129 296 21 90 7 552 97 252 17 502 197 9702 27 15 252 297 22 052 576 9 201 1 576 19 901 2 15 76 29 22 201 9 600 99 250 19 5550 199 9900 29 15 500 299 22 50 50 625 100 2500 150 5625 200 10 000 250 15 625 00 22 500 01 22 650 51 0 00 01 0 200 51 50 50 501 62 750 551 75 900 02 22 01 52 0 976 02 0 01 52 51 076 502 6 001 552 76 176 0 22 952 5 1 152 0 0 602 5 51 02 50 6 252 55 76 52 0 2 10 5 1 29 0 0 0 5 51 529 50 6 50 55 76 729 05 2 256 55 1 506 05 1 006 55 51 756 505 6 756 555 77 006 06 2 09 56 1 6 06 1 209 56 51 9 506 6 009 556 77 2 07 2 562 57 1 62 07 1 12 57 52 212 507 6 262 557 77 562 0 2 716 5 2 01 0 1 616 5 52 1 50 6 516 55 77 1 09 2 70 59 2 220 09 1 20 59 52 670 509 6 770 559 7 120 10 2 025 60 2 00 10 2 025 60 52 900 510 65 025 560 7 00 11 2 10 61 2 50 11 2 20 61 5 10 511 65 20 561 7 60 12 2 6 62 2 761 12 2 6 62 5 61 512 65 56 562 7 961 1 2 92 6 2 92 1 2 62 6 5 592 51 65 792 56 79 22 1 2 69 6 12 1 2 9 6 5 2 51 66 09 56 79 52 15 2 06 65 06 15 056 65 5 056 515 66 06 565 79 06 16 2 96 66 9 16 26 66 5 29 516 66 56 566 0 09 17 25 122 67 672 17 72 67 5 522 517 66 22 567 0 72 1 25 21 6 56 1 61 6 5 756 51 67 01 56 0 656 19 25 0 69 00 19 90 69 5 990 519 67 0 569 0 90 20 25 600 70 225 20 100 70 55 225 520 67 600 570 1 225 21 25 760 71 10 21 10 71 55 60 521 67 60 571 1 510 22 25 921 72 596 22 521 72 55 696 522 6 121 572 1 796 2 26 02 7 72 2 72 7 55 92 52 6 2 57 2 02 2 26 2 7 969 2 9 7 56 169 52 6 6 57 2 69 25 26 06 75 5 156 25 5 156 75 56 06 525 6 906 575 2 656 26 26 569 76 5 26 5 69 76 56 6 526 69 169 576 2 9 27 26 72 77 5 52 27 5 52 77 56 2 527 69 2 577 22 2 26 96 7 5 721 2 5 796 7 57 121 52 69 696 57 521 29 27 060 79 5 910 29 6 010 79 57 60 529 69 960 579 10 0 27 225 0 6 100 0 6 225 0 57 600 50 70 225 50 100 1 27 90 1 6 290 1 6 0 1 57 0 51 70 90 51 90 2 27 556 2 6 1 2 6 656 2 5 01 52 70 756 52 61 27 722 6 672 6 72 5 22 5 71 022 5 972 27 9 6 6 7 09 5 56 5 71 29 5 5 26 5 2 056 5 7 056 5 7 06 5 5 06 55 71 556 55 5 556 6 2 22 6 7 29 6 7 52 6 59 09 56 71 2 56 5 9 7 2 92 7 7 2 7 7 72 7 59 292 57 72 092 57 6 12 2 561 7 66 7 961 59 56 5 72 61 5 6 6 9 2 70 9 7 0 9 10 9 59 70 59 72 60 59 6 70 0 2 900 90 025 0 00 90 60 025 50 72 900 590 7 025 1 29 070 91 220 1 620 91 60 270 51 7 170 591 7 20 2 29 21 92 16 2 1 92 60 516 52 7 1 592 7 616 29 12 9 612 9 062 9 60 762 5 7 712 59 7 912 29 5 9 09 9 2 9 61 009 5 7 9 59 209 5 29 756 95 9 006 5 9 506 95 61 256 55 7 256 595 506 6 29 929 96 9 20 6 9 729 96 61 50 56 7 529 596 0 7 0 102 97 9 02 7 9 952 97 61 752 57 7 02 597 9 102 0 276 9 9 601 50 176 9 62 001 5 75 076 59 9 01 9 0 50 99 9 00 9 50 00 99 62 250 59 75 50 599 9 700 50 0 625 00 0 000 50 50 625 500 62 500 550 75 625 600 90 000 Frank Tapson 200 [trolfg:12]
Values of Triangle Numbers T n for n = 1 to 600 1 1 51 126 101 5151 151 11 76 201 20 01 251 1 626 2 52 17 102 525 152 11 62 202 20 50 252 1 7 6 5 11 10 556 15 11 71 20 20 706 25 2 11 10 5 15 10 560 15 11 95 20 20 910 25 2 5 5 15 55 150 105 5565 155 12 090 205 21 115 255 2 60 6 21 56 1596 106 5671 156 12 26 206 21 21 256 2 96 7 2 57 165 107 577 157 12 0 207 21 52 257 15 6 5 1711 10 56 15 12 561 20 21 76 25 11 9 5 59 1770 109 5995 159 12 720 209 21 95 259 670 10 55 60 10 110 6105 160 12 0 210 22 155 260 90 11 66 61 191 111 6216 161 1 01 211 22 66 261 191 12 7 62 195 112 62 162 1 20 212 22 57 262 5 1 91 6 2016 11 61 16 1 66 21 22 791 26 716 1 105 6 200 11 6555 16 1 50 21 2 005 26 90 15 120 65 215 115 6670 165 1 695 215 2 220 265 5 25 16 16 66 2211 116 676 166 1 61 216 2 6 266 5 511 17 15 67 227 117 690 167 1 02 217 2 65 267 5 77 1 171 6 26 11 7021 16 1 196 21 2 71 26 6 06 19 190 69 215 119 710 169 1 65 219 2 090 269 6 15 20 210 70 25 120 7260 170 1 55 220 2 10 270 6 55 21 21 71 2556 121 71 171 1 706 221 2 51 271 6 56 22 25 72 262 122 750 172 1 7 222 2 75 272 7 12 2 276 7 2701 12 7626 17 15 051 22 2 976 27 7 01 2 00 7 2775 12 7750 17 15 225 22 25 200 27 7 675 25 25 75 250 125 775 175 15 00 225 25 25 275 7 950 26 51 76 2926 126 001 176 15 576 226 25 651 276 226 27 7 77 00 127 12 177 15 75 227 25 7 277 50 2 06 7 01 12 256 17 15 91 22 26 106 27 71 29 5 79 160 129 5 179 16 110 229 26 5 279 9 060 0 65 0 20 10 515 10 16 290 20 26 565 20 9 0 1 96 1 21 11 66 11 16 71 21 26 796 21 9 621 2 52 2 0 12 77 12 16 65 22 27 02 22 9 90 561 6 1 911 1 16 6 2 27 261 2 0 16 595 570 1 905 1 17 020 2 27 95 2 0 70 5 60 5 655 15 910 15 17 205 25 27 70 25 0 755 6 666 6 71 16 916 16 17 91 26 27 966 26 1 01 7 70 7 2 17 95 17 17 57 27 2 20 27 1 2 71 916 1 9591 1 17 766 2 2 1 2 1 616 9 70 9 005 19 970 19 17 955 29 2 60 29 1 905 0 20 90 095 10 970 190 1 15 20 2 920 290 2 195 1 61 91 16 11 10 011 191 1 6 21 29 161 291 2 6 2 90 92 27 12 10 15 192 1 52 22 29 0 292 2 77 96 9 71 1 10 296 19 1 721 2 29 66 29 071 990 9 65 1 10 0 19 1 915 2 29 90 29 65 5 105 95 560 15 10 55 195 19 110 25 0 15 295 660 6 101 96 656 16 10 71 196 19 06 26 0 1 296 956 7 112 97 75 17 10 7 197 19 50 27 0 62 297 25 1176 9 51 1 11 026 19 19 701 2 0 76 29 551 9 1225 99 950 19 11 175 199 19 900 29 1 125 299 50 50 1275 100 5050 150 11 25 200 20 100 250 1 75 00 5 150 01 5 51 51 61 776 01 0 601 51 101 926 501 125 751 551 152 076 02 5 75 52 62 12 02 1 00 52 102 7 502 126 25 552 152 62 0 6 056 5 62 1 0 1 06 5 102 1 50 126 756 55 15 11 0 6 60 5 62 5 0 1 10 5 10 25 50 127 260 55 15 75 05 6 665 55 6 190 05 2 215 55 10 70 505 127 765 555 15 290 06 6 971 56 6 56 06 2 621 56 10 196 506 12 271 556 15 6 07 7 27 57 6 90 07 02 57 10 65 507 12 77 557 155 0 0 7 56 5 6 261 0 6 5 105 111 50 129 26 55 155 961 09 7 95 59 6 620 09 5 59 105 570 509 129 795 559 156 520 10 205 60 6 90 10 255 60 106 00 510 10 05 560 157 00 11 516 61 65 1 11 666 61 106 91 511 10 16 561 157 61 12 2 62 65 70 12 5 07 62 106 95 512 11 2 562 15 20 1 9 11 6 66 066 1 5 91 6 107 16 51 11 1 56 15 766 1 9 55 6 66 0 1 5 905 6 107 0 51 12 55 56 159 0 15 9 770 65 66 795 15 6 20 65 10 5 515 12 70 565 159 95 16 50 06 66 67 161 16 6 76 66 10 11 516 1 6 566 160 61 17 50 0 67 67 52 17 7 15 67 109 27 517 1 90 567 161 02 1 50 721 6 67 96 1 7 571 6 109 76 51 1 21 56 161 596 19 51 00 69 6 265 19 7 990 69 110 215 519 1 90 569 162 165 20 51 60 70 6 65 20 10 70 110 65 520 15 60 570 162 75 21 51 61 71 69 006 21 1 71 111 156 521 15 91 571 16 06 22 52 00 72 69 7 22 9 25 72 111 62 522 16 50 572 16 7 2 52 26 7 69 751 2 9 676 7 112 101 52 17 026 57 16 51 2 52 650 7 70 125 2 90 100 7 112 575 52 17 550 57 165 025 25 52 975 75 70 500 25 90 525 75 11 050 525 1 075 575 165 600 26 5 01 76 70 76 26 90 951 76 11 526 526 1 601 576 166 176 27 5 62 77 71 25 27 91 7 77 11 00 527 19 12 577 166 75 2 5 956 7 71 61 2 91 06 7 11 1 52 19 656 57 167 1 29 5 25 79 72 010 29 92 25 79 11 960 529 10 15 579 167 910 0 5 615 0 72 90 0 92 665 0 115 0 50 10 715 50 16 90 1 5 96 1 72 771 1 9 096 1 115 921 51 11 26 51 169 071 2 55 27 2 7 15 2 9 52 2 116 0 52 11 77 52 169 65 55 611 7 56 9 961 116 6 5 12 11 5 170 26 55 95 7 920 9 95 117 70 5 12 5 5 170 20 5 56 20 5 7 05 5 9 0 5 117 55 55 1 0 55 171 05 6 56 616 6 7 691 6 95 266 6 11 1 56 1 916 56 171 991 7 56 95 7 75 07 7 95 70 7 11 2 57 1 5 57 172 57 57 291 75 66 96 11 119 16 5 1 991 5 17 166 9 57 60 9 75 55 9 96 50 9 119 05 59 15 50 59 17 755 0 57 970 90 76 25 0 97 020 90 120 295 50 16 070 590 17 5 1 5 11 91 76 66 1 97 61 91 120 76 51 16 611 591 17 96 2 5 65 92 77 02 2 97 90 92 121 27 52 17 15 592 175 52 5 996 9 77 21 9 6 9 121 771 5 17 696 59 176 121 59 0 9 77 15 9 790 9 122 265 5 1 20 59 176 715 5 59 65 95 7 210 5 99 25 95 122 760 55 1 75 595 177 10 6 60 01 96 7 606 6 99 61 96 12 256 56 19 1 596 177 906 7 60 7 97 79 00 7 100 12 97 12 75 57 19 7 597 17 50 60 726 9 79 01 100 576 9 12 251 5 150 26 59 179 101 9 61 075 99 79 00 9 101 025 99 12 750 59 150 975 599 179 700 50 61 25 00 0 200 50 101 75 500 125 250 550 151 525 600 10 00 Frank Tapson 200 [trolfg:1]
Values of Square Numbers S n for n = 1 to 600 1 1 51 2601 101 10 201 151 22 01 201 0 01 251 6 001 01 90 601 51 12 201 01 160 01 51 20 01 501 251 001 551 0 601 2 52 270 102 10 0 152 2 10 202 0 0 252 6 50 02 91 20 52 12 90 02 161 60 52 20 0 502 252 00 552 0 70 9 5 209 10 10 609 15 2 09 20 1 209 25 6 009 0 91 09 5 12 609 0 162 09 5 205 209 50 25 009 55 05 09 16 5 2916 10 10 16 15 2 716 20 1 616 25 6 516 0 92 16 5 125 16 0 16 216 5 206 116 50 25 016 55 06 916 5 25 55 025 105 11 025 155 2 025 205 2 025 255 65 025 05 9 025 55 126 025 05 16 025 55 207 025 505 255 025 555 0 025 6 6 56 16 106 11 26 156 2 6 206 2 6 256 65 56 7 9 57 29 107 11 9 157 2 69 207 2 9 257 66 09 6 5 6 10 11 66 15 2 96 20 26 25 66 56 9 1 59 1 109 11 1 159 25 21 209 61 259 67 01 10 100 60 600 110 12 100 160 25 600 210 100 260 67 600 11 121 61 721 111 12 21 161 25 921 211 521 261 6 121 12 1 62 112 12 5 162 26 2 212 9 262 6 6 1 169 6 969 11 12 769 16 26 569 21 5 69 26 69 169 1 196 6 096 11 12 996 16 26 96 21 5 796 26 69 696 15 225 65 225 115 1 225 165 27 225 215 6 225 265 70 225 16 256 66 56 116 1 56 166 27 556 216 6 656 266 70 756 17 29 67 9 117 1 69 167 27 9 217 7 09 267 71 29 1 2 6 62 11 1 92 16 2 22 21 7 52 26 71 2 19 61 69 761 119 1 161 169 2 561 219 7 961 269 72 61 20 00 70 900 120 1 00 170 2 900 220 00 270 72 900 21 1 71 501 121 1 61 171 29 21 221 1 271 7 1 22 72 51 122 1 172 29 5 222 9 2 272 7 9 2 529 7 529 12 15 129 17 29 929 22 9 729 27 7 529 2 576 7 576 12 15 76 17 0 276 22 50 176 27 75 076 25 625 75 5625 125 15 625 175 0 625 225 50 625 275 75 625 26 676 76 5776 126 15 76 176 0 976 226 51 076 276 76 176 27 729 77 5929 127 16 129 177 1 29 227 51 529 277 76 729 2 7 7 60 12 16 17 1 6 22 51 9 27 77 2 29 1 79 621 129 16 61 179 2 01 229 52 1 279 77 1 0 900 0 600 10 16 900 10 2 00 20 52 900 20 7 00 1 961 1 6561 11 17 161 11 2 761 21 5 61 21 7 961 2 102 2 672 12 17 2 12 12 22 5 2 22 79 52 109 69 1 17 69 1 9 2 5 29 2 0 09 1156 7056 1 17 956 1 56 2 5 756 2 0 656 5 1225 5 7225 15 1 225 15 225 25 55 225 25 1 225 6 1296 6 796 16 1 96 16 596 26 55 696 26 1 796 7 169 7 7569 17 1 769 17 969 27 56 169 27 2 69 1 77 1 19 0 1 5 2 56 6 2 2 9 9 1521 9 7921 19 19 21 19 5 721 29 57 121 29 521 0 1600 90 100 10 19 600 190 6 100 20 57 600 290 100 1 161 91 21 11 19 1 191 6 1 21 5 01 291 61 2 176 92 6 12 20 16 192 6 6 22 5 56 292 5 26 19 9 69 1 20 9 19 7 29 2 59 09 29 5 9 196 9 6 1 20 76 19 7 66 2 59 56 29 6 6 5 2025 95 9025 15 21 025 195 025 25 60 025 295 7 025 6 2116 96 9216 16 21 16 196 16 26 60 516 296 7 616 7 2209 97 909 17 21 609 197 09 27 61 009 297 209 20 9 960 1 21 90 19 9 20 2 61 50 29 0 9 201 99 901 19 22 201 199 9 601 29 62 001 299 9 01 50 2500 100 10 000 150 22 500 200 0 000 250 62 500 00 90 000 06 9 66 56 126 76 06 16 6 56 207 96 506 256 06 556 09 16 07 9 29 57 127 9 07 165 69 57 20 9 507 257 09 557 10 29 0 9 6 5 12 16 0 166 6 5 209 76 50 25 06 55 11 6 09 95 1 59 12 1 09 167 21 59 210 61 509 259 01 559 12 1 10 96 100 60 129 600 10 16 100 60 211 600 510 260 100 560 1 600 11 96 721 61 10 21 11 16 921 61 212 521 511 261 121 561 1 721 12 97 62 11 0 12 169 7 62 21 512 262 1 562 15 1 97 969 6 11 769 1 170 569 6 21 69 51 26 169 56 16 969 1 9 596 6 12 96 1 171 96 6 215 296 51 26 196 56 1 096 15 99 225 65 1 225 15 172 225 65 216 225 515 265 225 565 19 225 16 99 56 66 1 956 16 17 056 66 217 156 516 266 256 566 20 56 17 100 9 67 1 69 17 17 9 67 21 09 517 267 29 567 21 9 1 101 12 6 15 2 1 17 72 6 219 02 51 26 2 56 22 62 19 101 761 69 16 161 19 175 561 69 219 961 519 269 61 569 2 761 20 102 00 70 16 900 20 176 00 70 220 900 520 270 00 570 2 900 21 10 01 71 17 61 21 177 21 71 221 1 521 271 1 571 26 01 22 10 6 72 1 22 17 0 72 222 7 522 272 572 27 1 2 10 29 7 19 129 2 17 929 7 22 729 52 27 529 57 2 29 2 10 976 7 19 76 2 179 776 7 22 676 52 27 576 57 29 76 25 105 625 75 10 625 25 10 625 75 225 625 525 275 625 575 0 625 26 106 276 76 11 76 26 11 76 76 226 576 526 276 676 576 1 776 27 106 929 77 12 129 27 12 29 77 227 529 527 277 729 577 2 929 2 107 5 7 12 2 1 1 7 22 52 27 7 57 0 29 10 21 79 1 61 29 1 01 79 229 1 529 279 1 579 5 21 0 10 900 0 1 00 0 1 900 0 20 00 50 20 900 50 6 00 1 109 561 1 15 161 1 15 761 1 21 61 51 21 961 51 7 561 2 110 22 2 15 92 2 16 62 2 22 2 52 2 02 52 72 110 9 16 69 17 9 2 29 5 2 09 5 9 9 111 556 17 56 1 56 2 256 5 25 156 5 1 056 5 112 225 5 1 225 5 19 225 5 25 225 55 26 225 55 2 225 6 112 96 6 1 996 6 190 096 6 26 196 56 27 296 56 96 7 11 569 7 19 769 7 190 969 7 27 169 57 2 69 57 569 11 2 150 5 191 2 1 5 29 5 5 7 9 11 921 9 151 21 9 192 721 9 29 121 59 290 521 59 6 921 0 115 600 90 152 100 0 19 600 90 20 100 50 291 600 590 100 1 116 21 91 152 1 1 19 1 91 21 01 51 292 61 591 9 21 2 116 96 92 15 66 2 195 6 92 22 06 52 29 76 592 50 6 117 69 9 15 9 196 29 9 2 09 5 29 9 59 51 69 11 6 9 155 26 197 16 9 2 06 5 295 96 59 52 6 5 119 025 95 156 025 5 19 025 95 25 025 55 297 025 595 5 025 6 119 716 96 156 16 6 19 916 96 26 016 56 29 116 596 55 216 7 120 09 97 157 609 7 199 09 97 27 009 57 299 209 597 56 09 121 10 9 15 0 200 70 9 2 00 5 00 0 59 57 60 9 121 01 99 159 201 9 201 601 99 29 001 59 01 01 599 5 01 50 122 500 00 160 000 50 202 500 500 250 000 550 02 500 600 60 000 Frank Tapson 200 [trolfg:1]
Values of Logarithms L n for n = 1.0 to 9.99 0 1 2 5 6 7 9 1.0 0.000 0.00 0.009 0.01 0.017 0.021 0.025 0.029 0.0 0.07 1.1 0.01 0.05 0.09 0.05 0.057 0.061 0.06 0.06 0.072 0.076 1.2 0.079 0.0 0.06 0.090 0.09 0.097 0.100 0.10 0.107 0.111 1. 0.11 0.117 0.121 0.12 0.127 0.10 0.1 0.17 0.10 0.1 1. 0.16 0.19 0.152 0.155 0.15 0.161 0.16 0.167 0.170 0.17 1.5 0.176 0.179 0.12 0.15 0.1 0.190 0.19 0.196 0.199 0.201 1.6 0.20 0.207 0.210 0.212 0.215 0.217 0.220 0.22 0.225 0.22 1.7 0.20 0.2 0.26 0.2 0.21 0.2 0.25 0.2 0.250 0.25 1. 0.255 0.25 0.260 0.262 0.265 0.267 0.270 0.272 0.27 0.276 1.9 0.279 0.21 0.2 0.26 0.2 0.290 0.292 0.29 0.297 0.299 2.0 0.01 0.0 0.05 0.0 0.10 0.12 0.1 0.16 0.1 0.20 2.1 0.22 0.2 0.26 0.2 0.0 0.2 0. 0.6 0. 0.0 2.2 0.2 0. 0.6 0. 0.50 0.52 0.5 0.56 0.5 0.60 2. 0.62 0.6 0.65 0.67 0.69 0.71 0.7 0.75 0.77 0.7 2. 0.0 0.2 0. 0.6 0.7 0.9 0.91 0.9 0.9 0.96 2.5 0.9 0.00 0.01 0.0 0.05 0.07 0.0 0.10 0.12 0.1 2.6 0.15 0.17 0.1 0.20 0.22 0.2 0.25 0.27 0.2 0.0 2.7 0.1 0. 0.5 0.6 0. 0.9 0.1 0.2 0. 0.6 2. 0.7 0.9 0.50 0.52 0.5 0.55 0.56 0.5 0.59 0.61 2.9 0.62 0.6 0.65 0.67 0.6 0.70 0.71 0.7 0.7 0.76.0 0.77 0.79 0.0 0.1 0. 0. 0.6 0.7 0.9 0.90.1 0.91 0.9 0.9 0.96 0.97 0.9 0.500 0.501 0.502 0.50.2 0.505 0.507 0.50 0.509 0.510 0.512 0.51 0.515 0.516 0.517. 0.519 0.520 0.521 0.522 0.52 0.525 0.526 0.52 0.529 0.50. 0.51 0.5 0.5 0.55 0.57 0.5 0.59 0.50 0.52 0.5.5 0.5 0.55 0.57 0.5 0.59 0.550 0.551 0.55 0.55 0.555.6 0.556 0.55 0.559 0.560 0.561 0.562 0.56 0.565 0.566 0.567.7 0.56 0.569 0.571 0.572 0.57 0.57 0.575 0.576 0.577 0.579. 0.50 0.51 0.52 0.5 0.5 0.55 0.57 0.5 0.59 0.590.9 0.591 0.592 0.59 0.59 0.596 0.597 0.59 0.599 0.600 0.601.0 0.602 0.60 0.60 0.605 0.606 0.607 0.609 0.610 0.611 0.612.1 0.61 0.61 0.615 0.616 0.617 0.61 0.619 0.620 0.621 0.622.2 0.62 0.62 0.625 0.626 0.627 0.62 0.629 0.60 0.61 0.62. 0.6 0.6 0.65 0.66 0.67 0.6 0.69 0.60 0.61 0.62. 0.6 0.6 0.65 0.66 0.67 0.6 0.69 0.650 0.651 0.652.5 0.65 0.65 0.655 0.656 0.657 0.65 0.659 0.660 0.661 0.662.6 0.66 0.66 0.665 0.666 0.667 0.667 0.66 0.669 0.670 0.671.7 0.672 0.67 0.67 0.675 0.676 0.677 0.67 0.679 0.679 0.60. 0.61 0.62 0.6 0.6 0.65 0.66 0.67 0.6 0.6 0.69.9 0.690 0.691 0.692 0.69 0.69 0.695 0.695 0.696 0.697 0.69 0 1 2 5 6 7 9 5.0 0.699 0.700 0.700 0.702 0.702 0.70 0.70 0.705 0.706 0.707 5.1 0.70 0.70 0.709 0.711 0.710 0.712 0.71 0.71 0.71 0.715 5.2 0.716 0.717 0.71 0.719 0.719 0.720 0.721 0.722 0.72 0.72 5. 0.72 0.725 0.726 0.727 0.72 0.72 0.729 0.70 0.71 0.72 5. 0.72 0.7 0.7 0.75 0.76 0.76 0.77 0.7 0.79 0.70 5.5 0.70 0.71 0.72 0.7 0.7 0.7 0.75 0.76 0.77 0.77 5.6 0.7 0.79 0.750 0.751 0.751 0.752 0.75 0.75 0.75 0.755 5.7 0.756 0.757 0.757 0.75 0.759 0.760 0.760 0.761 0.762 0.76 5. 0.76 0.76 0.765 0.766 0.766 0.767 0.76 0.769 0.769 0.770 5.9 0.771 0.772 0.772 0.77 0.77 0.775 0.775 0.776 0.777 0.777 6.0 0.77 0.779 0.70 0.70 0.71 0.72 0.72 0.7 0.7 0.75 6.1 0.75 0.76 0.77 0.77 0.7 0.79 0.790 0.790 0.791 0.792 6.2 0.792 0.79 0.79 0.79 0.795 0.796 0.797 0.797 0.79 0.799 6. 0.799 0.00 0.01 0.01 0.02 0.0 0.0 0.0 0.05 0.06 6. 0.06 0.07 0.0 0.0 0.09 0.10 0.10 0.11 0.12 0.12 6.5 0.1 0.1 0.1 0.15 0.16 0.16 0.17 0.1 0.1 0.19 6.6 0.20 0.20 0.21 0.22 0.22 0.2 0.2 0.2 0.25 0.25 6.7 0.26 0.27 0.27 0.2 0.29 0.29 0.0 0.1 0.1 0.2 6. 0. 0. 0. 0. 0.5 0.6 0.6 0.7 0. 0. 6.9 0.9 0.9 0.0 0.1 0.1 0.2 0. 0. 0. 0. 7.0 0.5 0.6 0.6 0.7 0. 0. 0.9 0.9 0.50 0.51 7.1 0.51 0.52 0.52 0.5 0.5 0.5 0.55 0.56 0.56 0.57 7.2 0.57 0.5 0.59 0.59 0.60 0.60 0.61 0.61 0.62 0.6 7. 0.6 0.6 0.65 0.65 0.66 0.66 0.67 0.67 0.6 0.69 7. 0.69 0.70 0.70 0.71 0.72 0.72 0.7 0.7 0.7 0.7 7.5 0.75 0.76 0.76 0.77 0.77 0.7 0.79 0.79 0.0 0.0 7.6 0.1 0.1 0.2 0. 0. 0. 0. 0.5 0.5 0.6 7.7 0.6 0.7 0. 0. 0.9 0.9 0.90 0.90 0.91 0.92 7. 0.92 0.9 0.9 0.9 0.9 0.95 0.95 0.96 0.97 0.97 7.9 0.9 0.9 0.99 0.99 0.900 0.900 0.901 0.901 0.902 0.90.0 0.90 0.90 0.90 0.905 0.905 0.906 0.906 0.907 0.907 0.90.1 0.90 0.909 0.910 0.910 0.911 0.911 0.912 0.912 0.91 0.91.2 0.91 0.91 0.915 0.915 0.916 0.916 0.917 0.91 0.91 0.919. 0.919 0.920 0.920 0.921 0.921 0.922 0.922 0.92 0.92 0.92. 0.92 0.925 0.925 0.926 0.926 0.927 0.927 0.92 0.92 0.929.5 0.929 0.90 0.90 0.91 0.91 0.92 0.92 0.9 0.9 0.9.6 0.95 0.95 0.96 0.96 0.97 0.97 0.9 0.9 0.99 0.99.7 0.90 0.90 0.91 0.91 0.92 0.92 0.9 0.9 0.9 0.9. 0.9 0.95 0.95 0.96 0.96 0.97 0.97 0.9 0.9 0.99.9 0.99 0.950 0.950 0.951 0.951 0.952 0.952 0.95 0.95 0.95 To find the value of the logarithm of n Match the first two digits of n (containing the decimal point) with those in the left-hand column of the table. Go along that line of the table to the three figure decimal number which lies directly under the value at the head of the table matching the third digit of n. That will be the logarithm of n. 9.0 0.95 0.955 0.955 0.956 0.956 0.957 0.957 0.95 0.95 0.959 9.1 0.959 0.960 0.960 0.960 0.961 0.961 0.962 0.962 0.96 0.96 9.2 0.96 0.96 0.965 0.965 0.966 0.966 0.967 0.967 0.96 0.96 9. 0.96 0.969 0.969 0.970 0.970 0.971 0.971 0.972 0.972 0.97 9. 0.97 0.97 0.97 0.975 0.975 0.975 0.976 0.976 0.977 0.977 9.5 0.97 0.97 0.979 0.979 0.90 0.90 0.90 0.91 0.91 0.92 9.6 0.92 0.9 0.9 0.9 0.9 0.95 0.95 0.95 0.96 0.96 9.7 0.97 0.97 0.9 0.9 0.99 0.99 0.99 0.990 0.990 0.991 9. 0.991 0.992 0.992 0.99 0.99 0.99 0.99 0.99 0.995 0.995 9.9 0.996 0.996 0.997 0.997 0.997 0.99 0.99 0.999 0.999 1.000 Frank Tapson 200 [trolfg:15]
1 2 5 6 7 9 1 1 2 5 6 7 9 2 2 6 10 12 1 16 1 6 9 12 15 1 21 2 27 12 16 20 2 2 2 6 5 5 10 15 20 25 0 5 0 5 6 6 12 1 2 0 6 2 5 7 7 1 21 2 5 2 9 56 6 16 2 2 0 56 6 72 9 9 1 27 6 5 5 6 72 1 1 2 5 6 7 9 1 1 2 5 6 7 9 2 2 6 10 12 1 16 1 6 9 12 15 1 21 2 27 12 16 20 2 2 2 6 5 5 10 15 20 25 0 5 0 5 6 6 12 1 2 0 6 2 5 7 7 1 21 2 5 2 9 56 6 16 2 2 0 56 6 72 9 9 1 27 6 5 5 6 72 1 1 2 5 6 7 9 1 1 2 5 6 7 9 2 2 6 10 12 1 16 1 6 9 12 15 1 21 2 27 12 16 20 2 2 2 6 5 5 10 15 20 25 0 5 0 5 6 6 12 1 2 0 6 2 5 7 7 1 21 2 5 2 9 56 6 16 2 2 0 56 6 72 9 9 1 27 6 5 5 6 72 1 1 2 5 6 7 9 1 1 2 5 6 7 9 2 2 6 10 12 1 16 1 6 9 12 15 1 21 2 27 12 16 20 2 2 2 6 5 5 10 15 20 25 0 5 0 5 6 6 12 1 2 0 6 2 5 7 7 1 21 2 5 2 9 56 6 16 2 2 0 56 6 72 9 9 1 27 6 5 5 6 72 1 1 2 5 6 7 9 1 1 2 5 6 7 9 2 2 6 10 12 1 16 1 6 9 12 15 1 21 2 27 12 16 20 2 2 2 6 5 5 10 15 20 25 0 5 0 5 6 6 12 1 2 0 6 2 5 7 7 1 21 2 5 2 9 56 6 16 2 2 0 56 6 72 9 9 1 27 6 5 5 6 72 1 1 2 5 6 7 9 1 1 2 5 6 7 9 2 2 6 10 12 1 16 1 6 9 12 15 1 21 2 27 12 16 20 2 2 2 6 5 5 10 15 20 25 0 5 0 5 6 6 12 1 2 0 6 2 5 7 7 1 21 2 5 2 9 56 6 16 2 2 0 56 6 72 9 9 1 27 6 5 5 6 72 1 1 2 5 6 7 9 1 1 2 5 6 7 9 2 2 6 10 12 1 16 1 6 9 12 15 1 21 2 27 12 16 20 2 2 2 6 5 5 10 15 20 25 0 5 0 5 6 6 12 1 2 0 6 2 5 7 7 1 21 2 5 2 9 56 6 16 2 2 0 56 6 72 9 9 1 27 6 5 5 6 72 1 1 2 5 6 7 9 1 1 2 5 6 7 9 2 2 6 10 12 1 16 1 6 9 12 15 1 21 2 27 12 16 20 2 2 2 6 5 5 10 15 20 25 0 5 0 5 6 6 12 1 2 0 6 2 5 7 7 1 21 2 5 2 9 56 6 16 2 2 0 56 6 72 9 9 1 27 6 5 5 6 72 1 Frank Tapson 200 [trolfg:16]
5 2 = 10 5 = 162 5 = 216 5 5 = 270 5 6 = 2 5 7 = 7 5 = 2 5 9 = 6 56 2 = 112 56 = 16 56 = 22 56 5 = 20 56 6 = 6 56 7 = 92 56 = 56 9 = 50 5 2 = 106 5 = 159 5 = 212 5 5 = 265 5 6 = 1 5 7 = 71 5 = 2 5 9 = 77 55 2 = 110 55 = 165 55 = 220 55 5 = 275 55 6 = 0 55 7 = 5 55 = 0 55 9 = 95 57 2 = 11 57 = 171 57 = 22 57 5 = 25 57 6 = 2 57 7 = 99 57 = 56 57 9 = 51 9 2 = 1 9 = 22 9 = 76 9 5 = 70 9 6 = 56 9 7 = 65 9 = 752 9 9 = 6 96 2 = 192 96 = 2 96 = 96 5 = 0 96 6 = 576 96 7 = 672 96 = 76 96 9 = 6 9 2 = 196 9 = 29 9 = 92 9 5 = 90 9 6 = 5 9 7 = 66 9 = 7 9 9 = 2 52 2 = 10 52 = 156 52 = 20 52 5 = 260 52 6 = 12 52 7 = 6 52 = 16 52 9 = 6 95 2 = 190 95 = 25 95 = 0 95 5 = 75 95 6 = 570 95 7 = 665 95 = 760 95 9 = 55 97 2 = 19 97 = 291 97 = 97 5 = 5 97 6 = 52 97 7 = 679 97 = 776 97 9 = 7 99 2 = 19 99 = 297 99 = 96 99 5 = 95 99 6 = 59 99 7 = 69 99 = 792 99 9 = 91 7 2 = 9 7 = 11 7 = 1 7 5 = 25 7 6 = 22 7 7 = 29 7 = 76 7 9 = 2 6 2 = 92 6 = 1 6 = 1 6 5 = 20 6 6 = 276 6 7 = 22 6 = 6 6 9 = 1 9 2 = 9 9 = 17 9 = 196 9 5 = 25 9 6 = 29 9 7 = 9 = 92 9 9 = 1 2 = 96 = 1 = 192 5 = 20 6 = 2 7 = 6 = 9 = 2 51 2 = 102 51 = 15 51 = 20 51 5 = 255 51 6 = 06 51 7 = 57 51 = 0 51 9 = 59 50 2 = 100 50 = 150 50 = 200 50 5 = 250 50 6 = 00 50 7 = 50 50 = 00 50 9 = 50 Multiplication Tables 1 2 5 6 7 9 1 1 2 5 6 7 9 2 2 6 10 12 1 16 1 6 9 12 15 1 21 2 27 12 16 20 2 2 2 6 5 5 10 15 20 25 0 5 0 5 6 6 12 1 2 0 6 2 5 7 7 1 21 2 5 2 9 56 6 16 2 2 0 56 6 72 9 9 1 27 6 5 5 6 72 1 70 2 = 10 70 = 210 70 = 20 70 5 = 50 70 6 = 20 70 7 = 90 70 = 560 70 9 = 60 72 2 = 1 72 = 216 72 = 2 72 5 = 60 72 6 = 2 72 7 = 50 72 = 576 72 9 = 6 7 2 = 1 7 = 222 7 = 296 7 5 = 70 7 6 = 7 7 = 51 7 = 592 7 9 = 666 71 2 = 12 71 = 21 71 = 2 71 5 = 55 71 6 = 26 71 7 = 97 71 = 56 71 9 = 69 7 2 = 16 7 = 219 7 = 292 7 5 = 65 7 6 = 7 7 = 511 7 = 5 7 9 = 657 75 2 = 150 75 = 225 75 = 00 75 5 = 75 75 6 = 50 75 7 = 525 75 = 600 75 9 = 675 2 2 = 56 2 = 2 = 112 2 5 = 10 2 6 = 16 2 7 = 196 2 = 22 2 9 = 252 0 2 = 60 0 = 90 0 = 120 0 5 = 150 0 6 = 10 0 7 = 210 0 = 20 0 9 = 270 2 2 = 6 2 = 96 2 = 12 2 5 = 160 2 6 = 192 2 7 = 22 2 = 256 2 9 = 2 29 2 = 5 29 = 7 29 = 116 29 5 = 15 29 6 = 17 29 7 = 20 29 = 22 29 9 = 261 1 2 = 62 1 = 9 1 = 12 1 5 = 155 1 6 = 16 1 7 = 217 1 = 2 1 9 = 279 2 = 66 = 99 = 12 5 = 165 6 = 19 7 = 21 = 26 9 = 297 22 2 = 22 = 66 22 = 22 5 = 110 22 6 = 12 22 7 = 15 22 = 176 22 9 = 19 2 2 = 2 = 72 2 = 96 2 5 = 120 2 6 = 1 2 7 = 16 2 = 192 2 9 = 216 26 2 = 52 26 = 7 26 = 10 26 5 = 10 26 6 = 156 26 7 = 12 26 = 20 26 9 = 2 2 2 = 6 2 = 69 2 = 92 2 5 = 115 2 6 = 1 2 7 = 161 2 = 1 2 9 = 207 25 2 = 50 25 = 75 25 = 100 25 5 = 125 25 6 = 150 25 7 = 175 25 = 200 25 9 = 225 27 2 = 5 27 = 1 27 = 10 27 5 = 15 27 6 = 162 27 7 = 19 27 = 216 27 9 = 2 76 2 = 152 76 = 22 76 = 0 76 5 = 0 76 6 = 56 76 7 = 52 76 = 60 76 9 = 6 7 2 = 156 7 = 2 7 = 12 7 5 = 90 7 6 = 6 7 7 = 56 7 = 62 7 9 = 702 0 2 = 160 0 = 20 0 = 20 0 5 = 00 0 6 = 0 0 7 = 560 0 = 60 0 9 = 720 77 2 = 15 77 = 21 77 = 0 77 5 = 5 77 6 = 62 77 7 = 59 77 = 616 77 9 = 69 79 2 = 15 79 = 27 79 = 16 79 5 = 95 79 6 = 7 79 7 = 55 79 = 62 79 9 = 711 1 2 = 162 1 = 2 1 = 2 1 5 = 05 1 6 = 6 1 7 = 567 1 = 6 1 9 = 729 Frank Tapson 200 [trolfg:17]
90 2 = 10 90 = 270 90 = 60 90 5 = 50 90 6 = 50 90 7 = 60 90 = 720 90 9 = 10 92 2 = 1 92 = 276 92 = 6 92 5 = 60 92 6 = 552 92 7 = 6 92 = 76 92 9 = 2 9 2 = 17 9 = 267 9 = 56 9 5 = 5 9 6 = 5 9 7 = 62 9 = 712 9 9 = 01 91 2 = 12 91 = 27 91 = 6 91 5 = 55 91 6 = 56 91 7 = 67 91 = 72 91 9 = 19 9 2 = 16 9 = 279 9 = 72 9 5 = 65 9 6 = 55 9 7 = 651 9 = 7 9 9 = 7 5 2 = 116 5 = 17 5 = 22 5 5 = 290 5 6 = 5 7 = 06 5 = 6 5 9 = 522 60 2 = 120 60 = 10 60 = 20 60 5 = 00 60 6 = 60 60 7 = 20 60 = 0 60 9 = 50 62 2 = 12 62 = 16 62 = 2 62 5 = 10 62 6 = 72 62 7 = 62 = 96 62 9 = 55 2 2 = 16 2 = 26 2 = 2 2 5 = 10 2 6 = 92 2 7 = 57 2 = 656 2 9 = 7 2 = 16 = 252 = 6 5 = 20 6 = 50 7 = 5 = 672 9 = 756 6 2 = 172 6 = 25 6 = 6 5 = 0 6 6 = 516 6 7 = 602 6 = 6 6 9 = 77 2 = 166 = 29 = 2 5 = 15 6 = 9 7 = 51 = 66 9 = 77 5 2 = 170 5 = 255 5 = 0 5 5 = 25 5 6 = 510 5 7 = 595 5 = 60 5 9 = 765 7 2 = 17 7 = 261 7 = 7 5 = 5 7 6 = 522 7 7 = 609 7 = 696 7 9 = 7 16 2 = 2 16 = 16 = 6 16 5 = 0 16 6 = 96 16 7 = 112 16 = 12 16 9 = 1 1 2 = 6 1 = 5 1 = 72 1 5 = 90 1 6 = 10 1 7 = 126 1 = 1 1 9 = 162 20 2 = 0 20 = 60 20 = 0 20 5 = 100 20 6 = 120 20 7 = 10 20 = 160 20 9 = 10 17 2 = 17 = 51 17 = 6 17 5 = 5 17 6 = 102 17 7 = 119 17 = 16 17 9 = 15 19 2 = 19 = 57 19 = 76 19 5 = 95 19 6 = 11 19 7 = 1 19 = 152 19 9 = 171 21 2 = 2 21 = 6 21 = 21 5 = 105 21 6 = 126 21 7 = 17 21 = 16 21 9 = 19 10 2 = 20 10 = 0 10 = 0 10 5 = 50 10 6 = 60 10 7 = 70 10 = 0 10 9 = 90 12 2 = 2 12 = 6 12 = 12 5 = 60 12 6 = 72 12 7 = 12 = 96 12 9 = 10 1 2 = 2 1 = 2 1 = 56 1 5 = 70 1 6 = 1 7 = 9 1 = 112 1 9 = 126 11 2 = 22 11 = 11 = 11 5 = 55 11 6 = 66 11 7 = 77 11 = 11 9 = 99 1 2 = 26 1 = 9 1 = 52 1 5 = 65 1 6 = 7 1 7 = 91 1 = 10 1 9 = 117 15 2 = 0 15 = 5 15 = 60 15 5 = 75 15 6 = 90 15 7 = 105 15 = 120 15 9 = 15 2 = 176 = 26 = 52 5 = 0 6 = 52 7 = 616 = 70 9 = 792 59 2 = 11 59 = 177 59 = 26 59 5 = 295 59 6 = 5 59 7 = 1 59 = 72 59 9 = 51 61 2 = 122 61 = 1 61 = 2 61 5 = 05 61 6 = 66 61 7 = 27 61 = 61 9 = 59 6 2 = 126 6 = 19 6 = 252 6 5 = 15 6 6 = 7 6 7 = 1 6 = 50 6 9 = 567 0 2 = 0 0 = 120 0 = 160 0 5 = 200 0 6 = 20 0 7 = 20 0 = 20 0 9 = 60 2 2 = 2 = 126 2 = 16 2 5 = 210 2 6 = 252 2 7 = 29 2 = 6 2 9 = 7 2 = = 12 = 176 5 = 220 6 = 26 7 = 0 = 52 9 = 96 1 2 = 2 1 = 12 1 = 16 1 5 = 205 1 6 = 26 1 7 = 27 1 = 2 1 9 = 69 2 = 6 = 129 = 172 5 = 215 6 = 25 7 = 01 = 9 = 7 5 2 = 90 5 = 15 5 = 10 5 5 = 225 5 6 = 270 5 7 = 15 5 = 60 5 9 = 05 2 = 6 = 102 = 16 5 = 170 6 = 20 7 = 2 = 272 9 = 06 6 2 = 72 6 = 10 6 = 1 6 5 = 10 6 6 = 216 6 7 = 252 6 = 2 6 9 = 2 2 = 76 = 11 = 152 5 = 190 6 = 22 7 = 266 = 0 9 = 2 5 2 = 70 5 = 105 5 = 10 5 5 = 175 5 6 = 210 5 7 = 25 5 = 20 5 9 = 15 7 2 = 7 7 = 111 7 = 1 7 5 = 15 7 6 = 222 7 7 = 259 7 = 296 7 9 = 9 2 = 7 9 = 117 9 = 156 9 5 = 195 9 6 = 2 9 7 = 27 9 = 12 9 9 = 51 6 2 = 12 6 = 192 6 = 256 6 5 = 20 6 6 = 6 7 = 6 = 512 6 9 = 576 66 2 = 12 66 = 19 66 = 26 66 5 = 0 66 6 = 96 66 7 = 62 66 = 52 66 9 = 59 6 2 = 16 6 = 20 6 = 272 6 5 = 0 6 6 = 0 6 7 = 76 6 = 5 6 9 = 612 65 2 = 10 65 = 195 65 = 260 65 5 = 25 65 6 = 90 65 7 = 55 65 = 520 65 9 = 55 67 2 = 1 67 = 201 67 = 26 67 5 = 5 67 6 = 02 67 7 = 69 67 = 56 67 9 = 60 69 2 = 1 69 = 207 69 = 276 69 5 = 5 69 6 = 1 69 7 = 69 = 552 69 9 = 621 Frank Tapson 200 [trolfg:1]
An alternative front cover for 'A Book of Multiplication Tables 10 to 99 Cut out and paste over the existing front cover, before making multiple copies. A Book of Multiplication Tables 10 to 99 Frank Tapson 200 [trolfg:19]